A Boundary Condition Capturing Method for Poisson's Equation on Irregular Domains

Li, Xudong; Fedkiw, Ronald; Kang, Myungjoo

doi:10.1006/jcph.2000.6444

Cited by 564 publications

(563 citation statements)

References 25 publications

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“…Standard finite differences cannot be applied across the interface due to the jump boundary conditions on Σ. The ghost cell method was developed to deal with this issue when solving elliptical problems by creating "ghost" computational points and using those ghost points in standard finite difference discretizations [15,[20][21][22]32]. In [34] and [36], we extended the ghost cell method to attain second-order accuracy on interior problems (i.e., p is constant in Ω c ) with boundary conditions that depend upon the geometry (e.g., curvature) and without a jump condition on the normal derivative.…”

Section: The Ghost Cell Methodsmentioning

confidence: 99%

“…However, by considering the normal derivative jump condition, we shall have two equations for p ℓ and p r , allowing us to completely eliminate them from the extrapolation. The proper discretization of the normal derivative jump [D∇p · n] across the interface has been an open problem since the introduction of the ghost cell method for the Poisson problem [15,21,32]. Suppose that we wish to discretize [D∇p · n] at the point x Σ = (x Σ , y j ) = (x i + θΔx, y j ) from the preceding discussion.…”

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

“…Suppose that we wish to discretize [D∇p · n] at the point x Σ = (x Σ , y j ) = (x i + θΔx, y j ) from the preceding discussion. In [32], the normal derivative jump was discretized as (28) where (29) Notice that this is equivalent to assuming that the normal vector cuts the computational mesh at a right angle (i.e., n = (1, 0)), potentially leading to numerical smearing of any jump in the tangential derivative. Furthermore, if θ ∼ 0 or θ ∼ 1, then the discretization can be unstable due to the uneven spacing of the stencil points.…”

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

“…4. In numerical testing in [32], this stencil was less than first-order accurate due to the low order of the discretization (first-order left and right differences of the derivative), coupled with the numerical smearing of the tangential derivatives.…”

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

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A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasisteady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics-an effect observed in real tumor growth.

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Section: The Ghost Cell Methodsmentioning

confidence: 99%

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

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A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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“…A full discussion of various fictitious domain methods for spectral/hp element methods is given by Vos et al (2008). Other methods have been described in the literature -see for example Liu et al (2000) and Hansbo and Hansbo (2002). The parameter " can be distributed at the quadrature points throughout the domain by the assignment of values depending on whether the region is solid or fluid, as follows:…”

Section: Fictitious Solid Obstaclesmentioning

confidence: 99%

A combined numerical and experimental framework for determining permeability properties of the arterial media

Comerford

Chooi

Nowak

et al. 2014

Biomech Model Mechanobiol

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The medial layer of the arterial wall may play an important role in the regulation of water and solute transport across the wall. In particular, a high medial resistance to transport could cause accumulation of lipid carrying molecules in the inner wall. In this study, the water transport properties of medial tissue were characterised in a numerical model utilising experimentally obtained data for the medial microstructure and the relative permeability of di↵erent constituents. For the model, a new solver for flow in porous materials, based on a high-order splitting scheme, was implemented in the spectral/hp element library nektar++ and validated. The data were obtained by immersing excised aortic bifurcations in a solution of fluorescent protein tracer and subsequently imaging them with a confocal microscope. Cuboidal regions of interest were selected in which the microstructure and relative permeability of di↵erent structures were transformed to a computational mesh. Impermeable objects were treated fictitiously in the numerical scheme. On this cube, a pressure drop was applied in the three coordinate directions and the principal components of the permeability tensor were determined. The reconstructed images demonstrated the arrangement of elastic lamellae and interspersed smooth muscle cells in rat aortic media; the distribution and alignment of the smooth muscle cells varied spatially within the extracellular matrix. The numerical simulations high-

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Physical Mechanism and Numerical Modeling of Liquid Propellant Atomization

2016

Internal Combustion Processes of Liquid Rocket Engines

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A Boundary Condition Capturing Method for Poisson's Equation on Irregular Domains

Cited by 564 publications

References 25 publications

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

A combined numerical and experimental framework for determining permeability properties of the arterial media

Physical Mechanism and Numerical Modeling of Liquid Propellant Atomization

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